How do you find the square root of 50?

##sqrt(50)## can be simplified as ##5sqrt(2)##

We can also find rational approximations to it.

For example:

##sqrt(50) ~~ 7 14/197 ~~ 7.071066##

The of ##50## is not a whole number, or even a rational number. It is an irrational number, but you can simplify it or find rational approximations for it.

First note that

##50 = 2 xx 5 xx 5##

contains a square factor ##5^2##. We can use this to simplify the square root:

##sqrt(50) = sqrt(5^2*2) = sqrt(5^2)*sqrt(2) = 5 sqrt(2)##

Apart from simplifying it algebraically, what is its numerical value?

Note that ##7^2 = 49##, so ##sqrt(49) = 7## and ##sqrt(50)## will be slightly larger than ##7##.

In fact, since ##50=7^2+1##, the square root of ##50## is expressible as a very regular continued fraction:

##sqrt(50) = 7+1/(14+1/(14+1/(14+1/(14+1/(14+1/(14+...))))))##

This can be written as ##sqrt(50) = [7;bar(14)]## where the bar over the ##14## indicates the repeating part of the continued fraction.

We can terminate the continued fraction early to give us rational approximations for ##sqrt(50)##.

For example:

##sqrt(50) ~~ [7;14] = 7+1/14 = 7.0bar(714285)##

##sqrt(50) ~~ [7;14,14] = 7+1/(14+1/14) = 7+14/197 ~~ 7.071066##

In fact:

##sqrt(50) ~~ 7.071067811865475244##